Categories in category theory

Category of small categories

In mathematics, specifically in category theory, the category of small categories, denoted by Cat, is the category whose objects are all small categories and whose morphisms are functors between categories. Cat may actually be regarded as a 2-category with natural transformations serving as 2-morphisms. The initial object of Cat is the empty category 0, which is the category of no objects and no morphisms. The terminal object is the terminal category or trivial category 1 with a single object and morphism. The category Cat is itself a large category, and therefore not an object of itself. In order to avoid problems analogous to Russell's paradox one cannot form the “category of all categories”. But it is possible to form a quasicategory (meaning objects and morphisms merely form a conglomerate) of all categories. (Wikipedia).

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Category Theory 1.2: What is a category?

What is a Category?

From playlist Category Theory

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Percentiles, Deciles, Quartiles

Understanding percentiles, quartiles, and deciles through definitions and examples

From playlist Unit 1: Descriptive Statistics

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Category Theory 2.1: Functions, epimorphisms

Functions, epimorphisms

From playlist Category Theory

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Category Theory 9.1: Natural transformations

Natural transformations

From playlist Category Theory

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Category Theory: The Beginner’s Introduction (Lesson 1 Video 4)

Lesson 1 is concerned with defining the category of Abstract Sets and Arbitrary Mappings. We also define our first Limit and Co-Limit: The Terminal Object, and the Initial Object. Other topics discussed include Duality and the Opposite (or Mirror) Category. These videos will be discussed

From playlist Category Theory: The Beginner’s Introduction

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Categories 6 Monoidal categories

This lecture is part of an online course on categories. We define strict monoidal categories, and then show how to relax the definition by introducing coherence conditions to define (non-strict) monoidal categories. We finish by defining symmetric monoidal categories and showing how super

From playlist Categories for the idle mathematician

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Higher Algebra 7: Non-abelian derived functors

In this video, we discuss the notion of non-abelian derived functors and Animation. Along the way, we discuss the Yoneda lemma. Warning: The Yoneda exercises stated at 35:00 is a bit hard given the technology we have, so I recommend simply proving the analogous statement for ordinary cat

From playlist Higher Algebra

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Category Theory: The Beginner’s Introduction (Lesson 1 Video 1)

Lesson 1 is concerned with defining the category of Abstract Sets and Arbitrary Mappings. We also define our first Limit and Co-Limit: The Terminal Object, and the Initial Object. Other topics discussed include Duality and the Opposite (or Mirror) Category. These videos will be discussed

From playlist Category Theory: The Beginner’s Introduction

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Category Theory 1.2 : Examples of Categories and Clarification

In this video, I clarify some terminology, and show some very important examples of categories. This includes the category of groups, sets, topologic spaces, monoids, modules, and rings. I also discuss the relation between categories and individual groups, preorders, matrices, and ordinals

From playlist Category Theory

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Emily Riehl: On the ∞-topos semantics of homotopy type theory: The simplicial model of...- Lecture 2

HYBRID EVENT Recorded during the meeting "Logic and Interactions" the February 22, 2022 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual M

From playlist Topology

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Camell Kachour - Globular perspective for Grothendieck ∞-topos and Grothendieck (∞,n)-topos

In this short talk we first briefly recall [4] how to build, for each integers n0, monads Tn on the category Glob of globular sets which algebras are globular models of (1; n)-categories, which have the virtue to be weak 1-categories of Penon and thus also to be weak 1-categories of Batani

From playlist Topos à l'IHES

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Julia Plavnik: "Classifying small fusion categories"

Actions of Tensor Categories on C*-algebras 2021 "Classifying small fusion categories" Julia Plavnik - Indiana University, Mathematics Abstract: Classifying fusion categories is a problem that at the moment seems out of reach, since it includes the classification of finite groups and sem

From playlist Actions of Tensor Categories on C*-algebras 2021

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Category Theory 9.2: bicategories

2-categories, bicategories

From playlist Category Theory

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Higher Algebra 1: ∞-Categories

In this video, we introduce ∞-categories. This is the first of a series of videos towards a reasonably non-technical overview over stable ∞-categories and Higher Algebra, which are intended to be watchable independently from the main lecture. Further resources: M.Boardman and R.Vogt. Homo

From playlist Higher Algebra

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Locally Cartesian Closed Infinity Categories - Joachim Kock

Joachim Kock Universitat Autonoma de Barcelona February 21, 2013 For more videos, visit http://video.ias.edu

From playlist Mathematics

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Emily Riehl: On the ∞-topos semantics of homotopy type theory: All ∞-toposes have... - Lecture 3

HYBRID EVENT Recorded during the meeting "Logic and Interactions" the February 24, 2022 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual M

From playlist Topology

Related pages

Complete category | Glossary of category theory | Functor | Initial and terminal objects | Forgetful functor | Functor category | Cartesian closed category | Free category | Mathematics | Exponential object | Russell's paradox | Conglomerate (mathematics) | Universal set | Category theory | Category (mathematics) | Natural transformation | Morphism